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It is shown that the drag of any semi-infinite airfoil section in purely subsonic inviscid flow follows precisely the Prandtl-Glauert compressibility rule. The result for the parabola has application to leading edge corrections in thin airfoil theory.
An investigation was made in the 5.18 m (17 ft) test section of the Langley 300 MPH 7 by 10 foot tunnel on a rectangular, aspect ratio 6 wing which had a slotted supercritical airfoil section and externally blown flaps. The 13 percent thick wing was fitted with two high lift flap systems: single slotted and double slotted. The designations single slotted and double slotted do not include the slot which exists near the trailing edge of the basic slotted supercritical airfoil. Tests were made over an angle of attack range of -6 deg to 20 deg and a thrust-coefficient range up to 1.94 for a free-stream dynamic pressure of 526.7 Pa (11.0 lb/sq ft). The results of the investigation are presented as curves and tabulations of the chordwise pressure distributions at the midsemispan station for the wing and each flap element.
A numerical technique for designing transonic airfoils having a prescribed pressure distribution (the inverse problem) is presented. The method employs the basic features of Jameson's iterative solution for the full potential equation, except that inverse boundary conditions and Cartesian coordinates are used. The method is a direct-inverse approach that controls trailing-edge closure. Examples show the application of the method to design aft-cambered and other airfoils specifically for transonic flight.
A practical procedure for the design of low drag, transonic airfoils is demonstrated. The procedure uses an optimization program, based on a gradient algorithm coupled with an aerodynamic analysis program, that solves the full, non-linear potential equation for transonic flow. The procedure is useful for the design of retrofit modifications for drag reduction of existing aircraft as well as for the design of low drag profiles for new aircraft. Results are presented for the modification of four different airfoils to decrease the drag at a given transonic Mach number.
An approximate numerical method has been developed for computing the entire transonic flow field over an airfoil, including the boundary layer, the shock-wave/boundary-layer interaction, and the shock-induced-separation. The inviscid flow, the turbulent boundary layer in the weak interaction zone, and the strong interaction elliptic region are solved separately. The inviscid flow is represented by the solution of the transonic small-disturbance equation, while the boundary layer is solved by the generalized Galerkin method, and the strong interaction region (including the wake) is computed from the time-averaged Navier-Stokes equations. A second-order interaction is computed iteratively using the inviscid pressure distribution as a boundary condition for the boundary layer and Navier-Stokes codes, and the viscous displacement thickness to modify the airfoil geometry for the inviscid computation. The computed pressure distributions and shock-wave profiles are compared with experimental data. Agreement is very good except that the pressure recovery on the trailing edge is greater than the experimental data.
Details of force, moment, and pressure distributions on a two-dimensional, 4-ft chord, NACA 0012 airfoil, oscillating in pitch through stall, in a 7 ft x 10 ft low-speed wind tunnel are presented. Tests were run with the airfoil in a closed test section and also in a test section having four longitudinal slots in each sidewall set to provide minimum tunnel interference on the wing in steady flow. In steady flow, differences between the results for the closed and 2% open case are small. The dynamic stall process is not triggered by the bursting of a laminar separation bubble but rather by the separation of the turbulent boundary layer downstream of the bubble.
There are plans to use the supercritical wing on the next generation of commercial aircraft so as to economize on fuel consumption by reducing drag. Computer codes have served well in meeting the consequent demand for new wing sections. The possibility of replacing wind tunnel tests by computational fluid dynamics is discussed. Another approach to the supercritical wing is through shockless airfoils. A novel boundary value problem in the hodograph plane is studied that enables one to design a shockless airfoil so that its pressure distribution very nearly takes on data that are prescribed.
A nonlinear procedure for computing the pressure distribution on an airfoil with a trailing edge jet flap is described. The method is not restricted to thin airfoils or shallow jet-deflection angles. Correlation with experiment indicates that the characteristics of the pressure distribution are predicted by the theory, but the effect of entrainment is overpredicted with the entrainment coefficient used.
Wind-tunnel tests were conducted to determine the low-speed section characteristics of a 13 percent-thick airfoil designed for general aviation applications. The results were compared with NACA 12 percent-thick sections and with the 17 percent-thick NASA airfoil. The tests were conducted ovar a Mach number range from 0.10 to 0.35. Chord Reynolds numbers varied from about 2,000,000 to 9,000,000.
A computer program was developed to solve the low speed flow around airfoils with highly separated flow. A new flow model included all of the major physical features in the separated region. Flow visualization tests also were made which gave substantiation to the validity of the model. The computation involves the matching of the potential flow, boundary layer and flows in the separated regions. Head's entrainment theory was used for boundary layer calculations and Korst's jet mixing analysis was used in the separated regions. A free stagnation point aft of the airfoil and a standing vortex in the separated region were modelled and computed.
A program called TRANDES is presented that is used for the analysis of steady, irrotational transonic flow over specified two-dimensional airfoils in free air or for the design of airfoils having a prescribed pressure distribution, including the effects of weak viscous interaction. Instructions on program usage, listings of the program, and sample cases are given.
Wind tunnel force and pressure tests were conducted for the GA(W)-1 airfoil equipped with a 20% aileron, and pressure tests were conducted with a 30% Fowler flap. All tests were conducted at a Reynolds number of 2.2 and a Mach number of 0.13. The aileron provides control effectiveness similar to ailerons applied to more conventional airfoils. Effects of aileron gaps from 0% to 2% chord were evaluated, as well as hinge moment characteristics. The aft camber of the GA(W)-1 section results in a substantial up-aileron moment, but the hinge moments associated with aileron deflection are similar to other configurations. Fowler flap pressure distributions indicate that unseparated flow is achieved for flap settings up to 40 deg., over a limited angle of attack range. Theoretical pressure distributions compare favorably with experiments for low flap deflections, but show substantial errors at large deflections.
A computer program is described for calculating the profile of an airfoil as well as the boundary layer momentum thickness and energy form parameter. The theory underlying the airfoil inversion technique developed by Eppler is discussed.
Measurements of surface pressure and wake flow fluctuations were made as part of a transonic wind tunnel investigation into the nature of a supercritical airfoil flow field. Emphasis was on a range of high subsonic Mach numbers and moderate lift coefficients corresponding to the development of drag divergence and buffeting. Fluctuation data were analyzed statistically for intensity, frequency content, and spatial coherence. Variations in these parameters were correlated with changes in the mean airfoil flow field.
A computer code, LTRAN2, has been constructed that efficiently computes low-frequency unsteady transonic flows about airfoils in motion. The code solves the two-dimensional, nonlinear, low-frequency, small-disturbance equation by an alternating-direction implicit (ADI) algorithm. First, as a check on the code, LTRAN2 is used to solve the linear, low-frequency, small-disturbance equation. The results are compared with known answers from linear theory. Then nonlinear results are presented. Tijdeman's experimental observations of shock wave motions resulting from airfoil flap oscillations are qualitatively reproduced by LTRAN2. These computations compare favorably with those of Magnus and Yoshihara and are obtained in substantially less computer time. Also the underlying theory of the governing equation and the construction of the algorithm are discussed.
Recent applications of numerical optimization to the design of advanced airfoils for transonic aircraft have shown that low-drag sections can be developed for a given design Mach number without an accompanying drag increase at lower Mach numbers. This is achieved by imposing a constraint on the drag coefficient at an off-design Mach number while the drag at the design Mach number is the objective function. Such a procedure doubles the computation time over that for single design-point problems, but the final result is worth the increased cost of computation. The ability to treat such multiple design-point problems by numerical optimization has been enhanced by the development of improved airfoil shape functions. Such functions permit a considerable increase in the range of profiles attainable during the optimization process.
This paper considers viscous flows with unseparated turbulent boundary layers over two-dimensional airfoils at transonic speeds. Conventional theoretical methods are based on boundary layer formulations which do not account for the effect of the curved wake and static pressure variations across the boundary layer in the trailing edge region. In this investigation an extended viscous theory is developed that accounts for both effects. The theory is based on a rational analysis of the strong turbulent interaction at airfoil trailing edges. The method of matched asymptotic expansions is employed to develop formal series solutions of the full Reynolds equations in the limit of Reynolds numbers tending to infinity. Procedures are developed for combining the local trailing edge solution with numerical methods for solving the full potential flow and boundary layer equations. Theoretical results indicate that conventional boundary layer methods account for only about 50% of the viscous effect on lift, the remaining contribution arising from wake curvature and normal pressure gradient effects.
Finite element procedures are presented for the numerical solution of transonic flow over lifting and oscillating airfoils based on small disturbance theory. The algorithm is simple and effective in handling the mixed flow problem and, for airfoil in free air, treats the entire flow domain by making use of the far field asymptotic solution. Numerical results are presented and compared with experimental data and those obtained by finite difference relaxation techniques.